Xenforo has a feature which gives an alert window within the browser / page when there is an active notification. It should be like check for notification at regular intervals and display a notification for the same. I found an Angular plugin which can give something like this
http://janstevens.github.io/angular-growl-2/
I personally find annoying even the PM notifications, so do not count on me for that one. :P
I don't want or need this too, sorry. I am refreshing the site so often (change pages back and forth, go to another board, etc) there isn't much time between these actions where a notification like this will come in handy.